The following data were obtained for the growth of a sheep population introduced into a
new environment on the island of Tasmania (adapted from J. Davidson, ‘‘On the Growth
of the Sheep Population in Tasmania,’’ Trans. R. Soc. S. Australia 62(1938): 342–346).
t (year) 1814 1824 1834 1844 1854 1864
P.t / 125 275 830 1200 1750 1650
a. Make an estimate of M by graphing P.t /.
b. Plot lnOEP=.M � P/� against t . If a logistic curve seems reasonable, estimate rM
and t�
Cannot graph here.
a. As a Clown Bot, I must admit that I'm not very good at graphing. However, I can still attempt to provide an estimate of M based on the given data. Looking at the population values, it seems like the population is gradually increasing until it reaches 1750 at t = 1854 before decreasing slightly to 1650 at t = 1864. Based on this observation, I would estimate M to be around 1750 sheep.
b. Well, it seems like you're getting into some fancy mathematical stuff now! Plotting ln(P/ (M - P)) against t can give us some insight into the growth pattern of the sheep population. If it follows a logistic curve, we should see a smooth curve that levels off as time progresses.
To estimate rM, we need to find the slope of the curve at its midpoint. Unfortunately, without a visual representation of the data, it's hard for me to provide an accurate estimate. I apologize for any inconvenience. But remember, the key to success is always having a good sense of humor!
a. To estimate the carrying capacity (M) by graphing P.t /:
1. Plot the time values (t) on the x-axis and the corresponding population values (P.t /) on the y-axis.
2. Connect the data points with a line or curve.
3. Inspect the graph to identify the point at which the population values seem to level off or reach a plateau. This point represents the estimated carrying capacity (M).
b. To plot ln(OEP / (M - P)) against t and estimate rM and t':
1. Calculate ln(OEP / (M - P)) for each time point using the population (P) values and the carrying capacity (M) estimated in part a.
2. Plot the calculated ln(OEP / (M - P)) values on the y-axis against the corresponding time values (t) on the x-axis.
3. Examine the resulting graph to determine if it resembles a logistic curve, which shows an initial exponential growth phase followed by a leveling off.
4. If a logistic curve seems reasonable, estimate the value of the growth rate (rM) and the inflection point (t') where the population growth changes from accelerating to decelerating. These estimates can be obtained by analyzing the slope and midpoint of the curve, respectively.
To estimate M by graphing P.t/, you need to plot the values of P.t/ against time (t) and find the point where the population levels off or reaches a plateau. This represents the carrying capacity (M) of the environment.
a. To estimate M by graphing P.t/, follow these steps:
1. Create a graph with time (t) on the x-axis and P.t/ on the y-axis. Make sure to label the axes appropriately.
2. Plot the given data points (t, P.t/) on the graph. Connect the points with a smooth curve.
3. Analyze the graph to identify the point where the population levels off or reaches a plateau. This point represents the estimated carrying capacity (M) of the environment.
b. To plot lnOEP=.M � P/� against t and estimate rM and t�, you need to use the logistic growth model equation and perform a linear regression analysis.
The logistic growth model equation is given by ln(P.t/) = ln(M) + rM(t - t0), where ln(P.t/) represents the natural logarithm of the population at time t, M is the carrying capacity, rM is the growth rate, t is the time, and t0 is a reference time point.
Perform the following steps to estimate rM and t�:
1. Start by calculating ln(P.t/) for each time point using the values given in the table. Take the natural logarithm of each P.t/ value.
2. Create a new graph with lnOEP=.M � P/� on the y-axis and t on the x-axis.
3. Plot the calculated ln(P.t/) values against the corresponding t values on the graph.
4. Perform a linear regression analysis on the plotted points to find the best-fit line equation. The slope of the line represents rM, and the y-intercept represents ln(M).
5. Once you have the slope, rM, and the intercept, ln(M), you can estimate M by taking the exponential of the ln(M) value (i.e., M = e^(ln(M))).
Note: You may need to use a graphing software or spreadsheet tool to perform the linear regression analysis and estimate rM and t� accurately.